I am stuck here. It looks like some sort of a binomial coefficient but I just can't figure out how wolfram alpha produces this exact solution:
$$\sum_{k=1}^\infty \frac{n}{(n+k)(n+k+1)}=\frac{n}{n+1}$$
2026-03-25 04:42:35.1774413755
Can someone explain to me how $\sum_{k=1}^\infty \frac{n}{(n+k)(n+k+1)}=\frac{n}{n+1}$?
80 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in SEQUENCES-AND-SERIES
- How to show that $k < m_1+2$?
- Justify an approximation of $\sum_{n=1}^\infty G_n/\binom{\frac{n}{2}+\frac{1}{2}}{\frac{n}{2}}$, where $G_n$ denotes the Gregory coefficients
- Negative Countdown
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Show that the sequence is bounded below 3
- A particular exercise on convergence of recursive sequence
- Proving whether function-series $f_n(x) = \frac{(-1)^nx}n$
- Powers of a simple matrix and Catalan numbers
- Convergence of a rational sequence to a irrational limit
- studying the convergence of a series:
Related Questions in ANALYSIS
- Analytical solution of a nonlinear ordinary differential equation
- Finding radius of convergence $\sum _{n=0}^{}(2+(-1)^n)^nz^n$
- Show that $d:\mathbb{C}\times\mathbb{C}\rightarrow[0,\infty[$ is a metric on $\mathbb{C}$.
- conformal mapping and rational function
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Proving whether function-series $f_n(x) = \frac{(-1)^nx}n$
- Elementary question on continuity and locally square integrability of a function
- Proving smoothness for a sequence of functions.
- How to prove that $E_P(\frac{dQ}{dP}|\mathcal{G})$ is not equal to $0$
- Integral of ratio of polynomial
Related Questions in SUMMATION
- Computing:$\sum_{n=0}^\infty\frac{3^n}{n!(n+3)}$
- Prove that $1+{1\over 1+{1\over 1+{1\over 1+{1\over 1+...}}}}=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}}$
- Fourier series. Find the sum $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n+1}$
- Sigma (sum) Problem
- How to prove the inequality $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n-1}\geq \log (2)$?
- Double-exponential sum (maybe it telescopes?)
- Simplify $\prod_{k=1}^{l} \sum_{r=d}^m {{m}\choose{r}} \left(N-k \right)^{r} k^{m-r+1}$
- Sum of two martingales
- How can we prove that $e^{-jωn}$ converges at $0$ while n -> infinity?
- Interesting inequalities
Related Questions in TELESCOPIC-SERIES
- Double-exponential sum (maybe it telescopes?)
- Calculate the sum of the series: $\sum\limits_{n=1}^{\infty}\left ( \frac{3}{n^{2}+n}-\frac{2}{4n^{2}+16n+15} \right )$
- Squared Summation(Intermediate Step)
- Another solution.
- I am stuck on a question Algebra:Sequence and series
- What is the closed form of the sequence : $ a_{n+2} = 2a_{n+1}-a_n+2^n+2 $
- Find the closed form of $u_{n+1}=a_nu_n+b_n$
- Want to understand how a fraction is simplified
- Evaluate a sum which almost looks telescoping but not quite:$\sum_{k=2}^n \frac{1}{k(k+2)}$
- Compute the values of the Double Sum
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
By the telescoping sum: $$\sum_{k=1}^{+\infty} \frac{n}{(n+k)(n+k+1)}=n\sum_{k=1}^{+\infty}\left(\frac{1}{n+k}-\frac{1}{n+k+1}\right)=$$ $$=n\lim_{k\rightarrow+\infty}\left(\frac{1}{n+1}-\frac{1}{n+k+1}\right)=\frac{n}{n+1}$$